We present observational evidence from studies on primary cortical cultures from AD transgenic mice, APPSwe/PS1ΔE9 (APP/PS1) mice, for significant decrease in total spine density at DIV-15 and onward. This indicates reduction in potential healthy synapses and strength of connections among neurons. Based on this, a network model of neurons is developed, that explains the consequent loss of coordinated activity and transmission efficiency among neurons that manifests over time. The critical time when structural connectivity in the brain undergoes a phase-transition, from initial robustness to irreparable breakdown, is estimated from this model. We also show how the global efficiency of signal transmission in the network decreases over time. Moreover, the number of multiple paths of high efficiency decreases rapidly as the disease progresses, indicating loss of structural plasticity and inefficiency in choosing alternate paths or desired paths for any pattern of activity. Thus loss of spines caused by β-Amyloid (Aβ) peptide results in disintegration of the neuronal network over time with consequent cognitive dysfunctions in Alzheimer's Disease (AD).